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We consider an asexual biological population of constant size N evolving in discrete time under the influence of selection and mutation. Beneficial mutations appear at rate U and their selective effects s are drawn from a distribution g(s). After introducing the required models and concepts of mathematical population genetics, we review different approaches to computing the speed of logarithmic fitness increase as a function of N, U and g(s). We present an exact solution of the infinite population size limit and provide an estimate of the population size beyond which it is valid. We then discuss approximate approaches to the finite population problem, distinguishing between the case of a single selection coefficient, g(s) = δ (s − s b ), and a continuous distribution of selection coefficients. Analytic estimates for the speed are compared to numerical simulations up to population sizes of order 10 300 .

We show that in the asymmetric simple exclusion process (ASEP) on a ring, conditioned on carrying a large flux, the particle experience an effective longrange potential which in the limit of very large flux takes the simple form U = −2 i =j log | sin π(n i /L − n j /L)|, where n 1 , n 2 , . . . n N are the particle positions, similar to the effective potential between the eigenvalues of the circular unitary ensemble in random matrices. Effective hopping rates and various quasistationary probabilities under such a conditioning are found analytically using the Bethe ansatz and determinantal free fermion techniques. Our asymptotic results extend to the limit of large current and large activity for a family of reaction-diffusion processes with onsite exclusion between particles. We point out an intriguing generic relation between classical stationary probability distributions for conditioned dynamics and quantum ground state wave functions, in particular, in the case of exclusion processes, for free fermions.

The asymmetric simple exclusion process with open boundaries, which is a very simple model of out-of-equilibrium statistical physics, is known to be integrable. In particular, its spectrum can be described in terms of Bethe roots. The large deviation function of the current can be obtained as well by diagonalizing a modified transition matrix, that is still integrable: the spectrum of this new matrix can be also described in terms of Bethe roots for special values of the parameters. However, due to the algebraic framework used to write the Bethe equations in the previous works, the nature of the excitations and the full structure of the eigenvectors were still unknown. This paper explains why the eigenvectors of the modified transition matrix are physically relevant, gives an explicit expression for the eigenvectors and applies it to the study of atypical currents. It also shows how the coordinate Bethe Ansatz developped for the excitations leads to a simple derivation of the Bethe equations and of the validity conditions of this Ansatz. All the results obtained by de Gier and Essler are recovered and the approach gives a physical interpretation of the exceptional points The overlap of this approach with other tools such as the matrix Ansatz is also discussed. The method that is presented here may be not specific to the asymmetric exclusion process and may be applied to other models with open boundaries to find similar exceptional points.

A branching random walk in presence of an absorbing wall moving at a constant velocity v undergoes a phase transition as v varies. The problem can be analyzed using the properties of the Fisher-Kolmogorov-Petrovsky-Piscounov (F-KPP) equation. We find that the survival probability of the branching random walk vanishes at a critical velocity vc of the wall with an essential singularity and we characterize the divergences of the relaxation times for v < vc and v > vc. At v = vc the survival probability decays like a stretched exponential. Using the F-KPP equation, one can also calculate the distribution of the population size at time t conditionned by the survival of one individual at a later time T > t. Our numerical results indicate that the size of the population diverges like the exponential of (vc − v) −1/2 in the quasi-stationary regime below vc. Moreover for v > vc, our data indicate that there is no quasi-stationary regime. * Electronic address: bernard.derrida@lps.ens.fr † Electronic address: damien.simon@lps.ens.fr

We present a generalization of the coordinate Bethe ansatz that allows us to solve integrable open XXZ and ASEP models with non-diagonal boundary matrices, provided their parameters obey some relations. These relations extend the ones already known in the literature in the context of algebraic or functional Bethe ansatz. The eigenvectors are represented as sums over cosets of the BC n Weyl group.

We present the construction of the full set of eigenvectors of the open ASEP and XXZ models with special constraints on the boundaries. The method combines both recent constructions of coordinate Bethe Ansatz and the old method of matrix Ansatz specific to the ASEP. This "matrix coordinate Bethe Ansatz" can be viewed as a noncommutative coordinate Bethe Ansatz, the non-commutative part being related to the algebra appearing in the matrix Ansatz.

By measuring or calculating coalescence times for several models of coalescence or evolution, with and without selection, we show that the ratios of these coalescence times become universal in the large size limit and we identify a few universality classes.